Properties

Degree $4$
Conductor $1279091$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 5·9-s + 11-s + 8·13-s − 4·16-s − 4·17-s − 7·25-s + 7·31-s − 12·43-s − 10·45-s + 16·47-s − 10·49-s + 2·55-s + 10·59-s + 24·61-s + 16·65-s − 14·67-s − 6·71-s + 8·73-s − 20·79-s − 8·80-s + 16·81-s − 12·83-s − 8·85-s − 14·97-s − 5·99-s − 32·103-s + ⋯
L(s)  = 1  + 0.894·5-s − 5/3·9-s + 0.301·11-s + 2.21·13-s − 16-s − 0.970·17-s − 7/5·25-s + 1.25·31-s − 1.82·43-s − 1.49·45-s + 2.33·47-s − 1.42·49-s + 0.269·55-s + 1.30·59-s + 3.07·61-s + 1.98·65-s − 1.71·67-s − 0.712·71-s + 0.936·73-s − 2.25·79-s − 0.894·80-s + 16/9·81-s − 1.31·83-s − 0.867·85-s − 1.42·97-s − 0.502·99-s − 3.15·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1279091 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1279091 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1279091\)    =    \(11^{3} \cdot 31^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{1279091} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1279091,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad11$C_1$ \( 1 - T \)
31$C_2$ \( 1 - 7 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.161876221686647141725563527465, −7.10826142797358908090370373880, −6.82697224947399950023698351410, −6.36261389471308870138602900888, −6.00363755168830900564762744465, −5.61864640510783414285306575446, −5.41935991651992802218472234441, −4.51842699512191534356663710538, −4.11715899745388408852378587541, −3.63170186460888703246647999593, −2.97562431456794399623394713691, −2.44672211519329826292315815599, −1.92484566977028663873402211272, −1.17041086963532389621044182318, 0, 1.17041086963532389621044182318, 1.92484566977028663873402211272, 2.44672211519329826292315815599, 2.97562431456794399623394713691, 3.63170186460888703246647999593, 4.11715899745388408852378587541, 4.51842699512191534356663710538, 5.41935991651992802218472234441, 5.61864640510783414285306575446, 6.00363755168830900564762744465, 6.36261389471308870138602900888, 6.82697224947399950023698351410, 7.10826142797358908090370373880, 8.161876221686647141725563527465

Graph of the $Z$-function along the critical line